First pass: the geometry is simple
In the first pass, we disregard the cylindrical shape of the world, and we assume that the light sources are in free space, moving against a black backdrop.
Let's assume that:
Each of the light sources moving through the tube produces the same amount of light as our own Sun; and
At midnight we want to have the same illumination as that produced by a full Moon.
Good to know:
The illumination produced by a full Moon (around 0.1 to 0.3 lux) is between 400,000 and 1,000,000 times weaker than the illumination produced by the Sun at noon (around 100,000 lux). (That's 19 to 20 exposure steps, in photographic terms.)
The illumination produced by a light source is inversely proportional to the square of the distance between the light source and the illuminated object.
With these assumptions, it follows that:
For the illumination produced by one of those moving light sources to decrease 800,000 to 2,000,000 times (the doubling is because we are illuminated by the next moving light source) it must move to a distance of 900 to 1400 astronomical units (= the radius of the orbit of the Earth, i.e., the radius of the cylinder assumed by the question).
The distance between two consecutive light sources will then be 1,800 to 2,800 astronomical units.
2000 a.u.<−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−> 1000 a.u.<−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−> \ | / Light source \ | / ··· --(•)-- ····················································· --(•)-- ···<<< / | \ <<< Movement ^ / | \ | | | 1 a.u.○ | Observer /|\ |Ground / \ v−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−////////////////////////////////////////////////////////////////////////////What about the speed of those light sources?
Hmm, that's a bummer. Light travels a distance of one astronomical unit in 8 minutes 20 seconds, which means that in one hour light travels a distance of 7.2 astronomical units, and in 12 hours it travels 86.4 astronomical units. Since the moving light sources need to travel about 1,000 astronomical units in 12 hours, it follows that they must move about 11.6 times faster than light.
Clearly, Einsteinian relativity doesn't apply in this world.
What the observer sees
At noon, the observer sees the light source overhead bathing the landscape in a sea of light, very very similar to what we see at noon.
At midnight, the observer sees a dark sky, with two very luminous stars at opposing points near the horizon.
Unlike on Earth, where the difference between daytime and nighttime is clear as day and night, on this world illumination varies gradually from full day to full night, with no clear separation between them. Most of the time it will be quite dark:
Illumination on a heavily clouded day is about 5 lux, or about 20,000 times lower than the illumination at noon on a clear day. Taking this as the threshold between day and twilight, the light source would have to be about 140 astronomical units distant, or one sixth of the 1,000 astronomical units which we considered midnight.
Taking the threshold between twilight and night to be 1 lux, that corresponds to a distance of some 320 astronomical units between the observer and the light source, or about 1/3 of the 1,000 astronomical units which we considered midnight.
All in all, in each 24 hours cycle, the observer will see about 4 hours daytime, about 16 hours night time, about 2 hours dawn and 2 hours twilight.
Second pass: but, but, but, but...
In the first pass we disregarded the cylindrical shape of the world, and we assumed that the light sources move against a black backdrop.
Now, that is perfectly fine as regards visible light. Assuming that the world has about the same albedo as Earth, the cylindrical shape of the world won't make a great difference. Yes, at noon there would be just a little more light than what the calculations in the first pass would suggest, etc. But the difference is utterly negligible, even for the keenest photographer.
The problem is not visible light, the problem is infrared light.
Earth likes very much to remain at constant temperature; see the great worldwide wailing at the prospect of increasing the average temperature by a measly one degree centigrade over a century.
Earth does this by radiating back into space all the energy it receives from the Sun. While the energy Earth receives from the Sun is mostly in the visible spectrum, the energy radiated by Earth is mostly in the infrared range.
And here comes the catch: those infinitely many sources of light will make the inner surface of the cylinder as hot as the Sun in a very short time. (Short time, geologically speaking, of course.)
Let's see what happens with a random square meter of ground in this cylindrical world:
During daytime, that square meter of ground is warmed up by the visible light falling on it.
At night, on our spherical Earth, that square meter of ground emits the heat in the form of infrared light. Most of the infrared energy is lost into outer space; some of it warms the air a little, and is then re-emitted by the air in the form of far infrared. Eventually, all the thermal energy dissipated by the square meter of ground as infrared radiation is lost into outer space.
But on this cylindrical world there is no outer space. All the energy ever received by that square meter of ground remains in the system forever. At night, the square meter of ground emits infrared light, but it does not help, because it absorbs the same amount of infrared light emitted by all those other square meters of land elsewhere on the inner surface of the cylinder.
Every 24 hours more and more energy is added to the square meter of ground, and it has nowhere to go except to warm up other square meters of ground. In a short time, every square meter of ground on the inner surface of the cylinder will be in thermal equilibrium with the sources of energy.